An elliptic-parabolic equation with a nonlocal term for the transient regime of a plasma in a Stellarator

We prove the existence and the regularity of weak solutions of a nonlocal elliptic-parabolic free-boundary problem involving the notions of relative rearrangement and monotone rearrangement. The problem arises in the study of the dynamics of a magnetically confined fusion plasma in a Stellarator device when the dimensional analysis on the characteristic times suggests to neglect the inertial acceleration in presence of a time dependent magnetic field.     

A mixed problem with Tricomi and Frankl conditions for the gellerstedt equation with a singular coefficient

We consider a generalized Tricomi equation with a singular coefficient. For this equation in a mixed domain we study the corresponding problem in the case, when a part of the boundary characteristic is free of boundary conditions; the deficient Tricomi condition is equivalently substituted by a nonlocal Frankl condition on a segment of the degeneration line. We prove that the stated problem is well-posed.     

A two-phase flow with a viscous and an inviscid fluid

We study the free boundary between a viscous and an inviscid fluid satisfying the Navier-Stokes and Euler equations respectively. Surface tension is incorporated. We read the equations as a free boundary problem for one viscous fluid with a nonlocal boundary force. We decompose the pressure distribution in the inviscid fluid into two contributions. A positivity result for the first, and a compactness property for the second contribution are dervied. We prove a short time existence theorem for the two-phase problem     

Singular Perturbation in a Nonlocal Diffusion Problem

We study a singular perturbation problem for a nonlocal evolution operator. The problem appears in the analysis of the propagation of flames in the high activation energy limit, when admitting nonlocal effects.

We obtain uniform estimates and we show that, under suitable assumptions, limits are solutions to a free boundary problem in a viscosity sense and in a pointwise sense at regular free boundary points.

We study the nonlocal problem both for a single equation and for a system of two equations.

Some of the results obtained are new even when the operator under consideration is the heat operator.     

On a nonlocal problem for a confined plasma in a Tokamak

The paper deals with a nonlocal problem related to the equilibrium of a confined plasma in a Tokamak machine. This problem involves terms u′_* (|u > u(x)|) and |u > u(x)|, which are neither local, nor continuous, nor monotone. By using the Galerkin approximate method and establishing some properties of the decreasing rearrangement, we prove the existence of solutions to such problem.     

Regularity of solutions to a free boundary problem modeling tumor growth by Stokes equation

In this paper we investigate regularity of solutions to a free boundary problem modeling tumor growth in fluid-like tissues. The model equations include a quasi-stationary diffusion equation for the nutrient concentration, and a Stokes equation with a source representing the proliferation density of the tumor cells, subject to a boundary condition with stress tensor effected by surface tension. This problem is a fully nonlinear problem involving nonlocal terms. Based on the employment of the functional analytic method and the theory of maximal regularity, we prove that the free boundary of this problem is real analytic in temporal and spatial variables for initial data of less regularity.     

Inverse problem for an equation of parabolic-hyperbolic type with a nonlocal boundary condition

For an equation of the parabolic-hyperbolic type, we consider an inverse problem with a nonlocal condition relating solution derivatives that belong to different types of the equation in question. We justify a uniqueness criterion and prove the existence of a solution of the problem by the spectral analysis method. We prove the stability of the solution with respect to the nonlocal boundary condition.     

Criterion for the uniqueness of a solution of a nonlocal problem for a degenerate equation of the mixed type in a rectangular domain

For a degenerate equation of the mixed type in a rectangular domain, we pose a nonlocal boundary value problem. By the spectral analysis method, we prove a criterion for the uniqueness of the solution of the problem.     

Dynamics for a nonlocal competition system with a free boundary

In this paper, we investigate a nonlocal reaction–diffusion competition model with a free boundary and discuss the long time behavior of species. The main objective is to understand the effect of the nonlocal term in the form of an integral convolution on the dynamics of competing species. Specially, for the weak competition case, when spreading occurs, we provide some sufficient conditions to prove that two competing species stabilize at a positive constant equilibrium state. Furthermore, for the case of successful spreading, we estimate the asymptotic spreading speed of the free boundary.     

Locally one-dimensional difference scheme for a pseudoparabolic equation with nonlocal conditions

We study a two-dimensional linear pseudoparabolic equation with nonlocal integral boundary conditions in one coordinate direction and use a locally one-dimensional method for solving this problem. We prove the stability of a finite-difference scheme based on the structure of spectrum of the difference operator with nonlocal conditions.